Abstract
This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a signchanging continuous function and may be unbounded from below.
1 Introduction
Consider the following SturmLiouville problem with integral boundary conditions
where
Problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3]. Integral BCs (BCs denotes boundary conditions) cover multipoint BCs and nonlocal BCs as special cases and have attracted great attention, see [414] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [4], Agarwal and O’Regan [5]. Yang [6], Boucherif [8], Chamberlain et al.[10], Feng [11], Jiang et al.[14] focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated twopoint and multipoint boundary value problems with signchanging nonlinear terms, for example, [1520], results for boundary value problems with integral boundary conditions when the nonlinear term is signchanging are rarely seen except for a few special cases [7,12,13].
Inspired by the above papers, the aim of this paper is to establish the existence of nontrivial solutions to BVP (1.1) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear term f of BVP (1.1) is allowed to be signchanging and unbounded from below. Secondly, the boundary conditions in BVP (1.1) are the RiemannStieltjes integral, which includes multipoint boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear operator. This paper employs different conditions and different methods to solve the same BVP (1.1) as [7]; meanwhile, this paper generalizes the result in [17] to boundary value problems with integral boundary conditions. What we obtain here is different from [620].
2 Preliminaries and lemmas
Let
We assume that the following condition holds throughout this paper.
(H_{1})
Let
Let
(H_{2})
Lemma 2.1 ([7])
If (H_{1}) and (H_{2}) hold, then BVP (1.1) is equivalent to
where
Define an operator
It is easy to show that
For any
It is easy to show that
and
Here
The representation of
Lemma 2.2 ([7])
If (H_{1}) holds, then there is
Lemma 2.3 ([22])
LetEbe a real Banach space and
Lemma 2.4Assume that (H_{1}), (H_{2}) and the following assumptions are satisfied:
(C_{1}) There exist
(C_{2}) There exists a continuous operator
(C_{3}) There exist a bounded continuous operator
(C_{4}) There exist
Let
where
Proof Choose a constant
Now we shall show
provided that R is sufficiently large.
In fact, if (2.7) is not true, then there exist
Since
Thus,
By (2.9),
where
(C_{3}) shows
The definition of
It follows from (2.6), (2.11) and (2.12) that
where
Since
□
3 Main results
Theorem 3.1Assume that (H_{1}), (H_{2}) hold and the following conditions are satisfied:
(A_{1}) There exist two nonnegative functions
(A_{2})
(A_{3})
(A_{4})
Here
Then BVP (1.1) has at least one nontrivial solution.
Proof We first show that all the conditions in Lemma 2.4 are satisfied. By Lemma 2.2, condition
(C_{1}) of Lemma 2.4 is satisfied. Obviously,
that is,
Take
which shows that condition (C_{3}) in Lemma 2.4 holds.
By (A_{3}), there exist
Combining (3.1) with (A_{1}), there exists
and so
Since K is a positive linear operator, from (3.2) we have
So condition (C_{4}) in Lemma 2.4 is satisfied.
According to Lemma 2.4, we derive that there exists a sufficiently large number
From (A_{4}) it follows that there exist
Thus
Next we will prove that
If there exist
It follows from (3.5) and Lemma 2.3 that
By (3.3), (3.6) and the additivity of LeraySchauder degree, we obtain
So A has at least one fixed point on
Corollary 3.1Using (
(
Competing interests
The authors declare that no conflict of interest exists.
Authors’ contributions
All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
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